E EP band and the HW band give narrower intervals for the middle portion of the data range than the piecewise Cox model. Near the end of the data range, all 3 bands have about the same width as the confidence interval from Prentice and others (2005). Overall the EP band matches most closely with the results for the piecewise constant hazards ratio model. The width of the EP band isFig. 1. The 95 pointwise confidence intervals and simultaneous confidence bands of the Actidione cancer hazard ratio function for the WHI data: Solid lines–equal precision confidence band; dashed lines– Hall ellner type confidence band; dash dotted lines– unweighted confidence band; outside dotted lines–pointwise confidence limits; and central dotted line–the estimated hazard ratio function; vertical segments–95 confidence intervals for the hazard ratios in 0?, 2?, and > years intervals from Prentice and others (2005), over the median of uncensored data in each time interval, 5 with “” indicating the point estimates.Estimation of the 2-sample hazard ratio function using a semiparametric modelless than or equal to the piecewise model ased confidence intervals for most of the data range, except at the beginning. Note that the constant function 1 is not excluded in the HW and UW bands. In comparison, the EP band stays above 1 for about the first 600 days. From Prentice and others (2005), the confidence interval for 0 – 2yr excludes 1, indicating an T0901317 biological activity elevation in coronary heart disease risk for the treatment early on. For this data set, the standard error of the estimated hazard ratio begins at 0.43, quickly comes down to below 0.20 at 600 days and stays below 0.20 for the rest of data range. Since the UW band does not take the variance into account and the HW band emphasizes the middle range, the elevated standard error at early follow-up times likely explains the discrepancy among the results. Compared with the original analysis that showed an overall difference between the 2 groups, the results here and those from Prentice and others (2005) give more detailed analysis on the dependence of the hazard ratio on time and help explaining the discrepancy between the results of the WHI clinical trial and preceding observational research, much of which involved cohorts where women could be enrolled some years after initiating hormone therapy. For the average hazard ratio function, the estimator and the 95 simultaneous confidence band are given in Figure 2. The standard error of the estimated average hazard ratio varies more mildly over time, and both the estimated average hazard ratio and the confidence band are changing much more smoothly compared with the results for the hazard ratio in Figure 1. Note that the confidence band stays above 1 for t < 700 days. This is in agreement with the results of Prentice and others (2005). To compare with the nonparametric approach, Figure 3 gives the estimated hazard ratio, the 95 pointwise confidence intervals and simultaneous confidence band of Gilbert and others (2002), based on the R programs from the author's site. The same scale as that in Figure 1 is used for comparison and results in truncation of some portion of the plot. The estimated hazard ratio suggests that the hazard ratio is reasonably monotonic. The nonparametric hazard ratio estimate is somewhat lower than the hazard ratio estimates in Figure 1 under either model (2.1) or the piecewise constant hazards ratio model. The confidence band is wider than those in Figure 1 for.E EP band and the HW band give narrower intervals for the middle portion of the data range than the piecewise Cox model. Near the end of the data range, all 3 bands have about the same width as the confidence interval from Prentice and others (2005). Overall the EP band matches most closely with the results for the piecewise constant hazards ratio model. The width of the EP band isFig. 1. The 95 pointwise confidence intervals and simultaneous confidence bands of the hazard ratio function for the WHI data: Solid lines--equal precision confidence band; dashed lines-- Hall ellner type confidence band; dash dotted lines-- unweighted confidence band; outside dotted lines--pointwise confidence limits; and central dotted line--the estimated hazard ratio function; vertical segments--95 confidence intervals for the hazard ratios in 0?, 2?, and > years intervals from Prentice and others (2005), over the median of uncensored data in each time interval, 5 with “” indicating the point estimates.Estimation of the 2-sample hazard ratio function using a semiparametric modelless than or equal to the piecewise model ased confidence intervals for most of the data range, except at the beginning. Note that the constant function 1 is not excluded in the HW and UW bands. In comparison, the EP band stays above 1 for about the first 600 days. From Prentice and others (2005), the confidence interval for 0 – 2yr excludes 1, indicating an elevation in coronary heart disease risk for the treatment early on. For this data set, the standard error of the estimated hazard ratio begins at 0.43, quickly comes down to below 0.20 at 600 days and stays below 0.20 for the rest of data range. Since the UW band does not take the variance into account and the HW band emphasizes the middle range, the elevated standard error at early follow-up times likely explains the discrepancy among the results. Compared with the original analysis that showed an overall difference between the 2 groups, the results here and those from Prentice and others (2005) give more detailed analysis on the dependence of the hazard ratio on time and help explaining the discrepancy between the results of the WHI clinical trial and preceding observational research, much of which involved cohorts where women could be enrolled some years after initiating hormone therapy. For the average hazard ratio function, the estimator and the 95 simultaneous confidence band are given in Figure 2. The standard error of the estimated average hazard ratio varies more mildly over time, and both the estimated average hazard ratio and the confidence band are changing much more smoothly compared with the results for the hazard ratio in Figure 1. Note that the confidence band stays above 1 for t < 700 days. This is in agreement with the results of Prentice and others (2005). To compare with the nonparametric approach, Figure 3 gives the estimated hazard ratio, the 95 pointwise confidence intervals and simultaneous confidence band of Gilbert and others (2002), based on the R programs from the author's site. The same scale as that in Figure 1 is used for comparison and results in truncation of some portion of the plot. The estimated hazard ratio suggests that the hazard ratio is reasonably monotonic. The nonparametric hazard ratio estimate is somewhat lower than the hazard ratio estimates in Figure 1 under either model (2.1) or the piecewise constant hazards ratio model. The confidence band is wider than those in Figure 1 for.
